| Label |
Class |
Class size |
Class degree |
Base field |
Field degree |
Field signature |
Conductor |
Conductor norm |
Discriminant norm |
Root analytic conductor |
Bad primes |
Rank |
Torsion |
CM |
CM |
Sato-Tate |
$\Q$-curve |
Base change |
Semistable |
Potentially good |
Nonmax $\ell$ |
mod-$\ell$ images |
$Ш_{\textrm{an}}$ |
Tamagawa |
Regulator |
Period |
Leading coeff |
j-invariant |
Weierstrass coefficients |
Weierstrass equation |
| 1369.1-a1 |
1369.1-a |
$3$ |
$9$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
1369.1 |
\( 37^{2} \) |
\( 37^{6} \) |
$2.49086$ |
$(a+6), (a-7)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3Cs.1.1 |
$1$ |
\( 3^{2} \) |
$1$ |
$4.739517032$ |
1.034247407 |
\( \frac{1404928000}{50653} \) |
\( \bigl[0\) , \( 1\) , \( 1\) , \( -23\) , \( -50\bigr] \) |
${y}^2+{y}={x}^{3}+{x}^{2}-23{x}-50$ |
| 1369.1-a2 |
1369.1-a |
$3$ |
$9$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
1369.1 |
\( 37^{2} \) |
\( 37^{2} \) |
$2.49086$ |
$(a+6), (a-7)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 1 \) |
$1$ |
$42.65565329$ |
1.034247407 |
\( \frac{4096000}{37} \) |
\( \bigl[0\) , \( 1\) , \( 1\) , \( -3\) , \( 1\bigr] \) |
${y}^2+{y}={x}^{3}+{x}^{2}-3{x}+1$ |
| 1369.1-a3 |
1369.1-a |
$3$ |
$9$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
1369.1 |
\( 37^{2} \) |
\( 37^{2} \) |
$2.49086$ |
$(a+6), (a-7)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3B.1.2 |
$9$ |
\( 1 \) |
$1$ |
$0.526613003$ |
1.034247407 |
\( \frac{727057727488000}{37} \) |
\( \bigl[0\) , \( 1\) , \( 1\) , \( -1873\) , \( -31833\bigr] \) |
${y}^2+{y}={x}^{3}+{x}^{2}-1873{x}-31833$ |
| 1369.1-b1 |
1369.1-b |
$3$ |
$9$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
1369.1 |
\( 37^{2} \) |
\( 37^{6} \) |
$2.49086$ |
$(a+6), (a-7)$ |
$2$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3Cs.1.1 |
$1$ |
\( 3^{2} \) |
$0.286221965$ |
$12.49778992$ |
3.122385515 |
\( \frac{1404928000}{50653} \) |
\( \bigl[0\) , \( a\) , \( 1\) , \( 117 a - 325\) , \( -1078 a + 3009\bigr] \) |
${y}^2+{y}={x}^{3}+a{x}^{2}+\left(117a-325\right){x}-1078a+3009$ |
| 1369.1-b2 |
1369.1-b |
$3$ |
$9$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
1369.1 |
\( 37^{2} \) |
\( 37^{2} \) |
$2.49086$ |
$(a+6), (a-7)$ |
$2$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3B.1.2 |
$1$ |
\( 1 \) |
$0.286221965$ |
$12.49778992$ |
3.122385515 |
\( \frac{4096000}{37} \) |
\( \bigl[0\) , \( a\) , \( 1\) , \( 17 a - 45\) , \( 46 a - 128\bigr] \) |
${y}^2+{y}={x}^{3}+a{x}^{2}+\left(17a-45\right){x}+46a-128$ |
| 1369.1-b3 |
1369.1-b |
$3$ |
$9$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
1369.1 |
\( 37^{2} \) |
\( 37^{2} \) |
$2.49086$ |
$(a+6), (a-7)$ |
$2$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 1 \) |
$2.575997686$ |
$12.49778992$ |
3.122385515 |
\( \frac{727057727488000}{37} \) |
\( \bigl[0\) , \( -a + 1\) , \( 1\) , \( -9367 a - 16858\) , \( 754620 a + 1351950\bigr] \) |
${y}^2+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-9367a-16858\right){x}+754620a+1351950$ |
| 1369.1-c1 |
1369.1-c |
$1$ |
$1$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
1369.1 |
\( 37^{2} \) |
\( 37^{2} \) |
$2.49086$ |
$(a+6), (a-7)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$24.03723960$ |
5.245355714 |
\( \frac{110592}{37} \) |
\( \bigl[0\) , \( 0\) , \( 1\) , \( 5 a - 14\) , \( 6 a - 17\bigr] \) |
${y}^2+{y}={x}^{3}+\left(5a-14\right){x}+6a-17$ |
| 1369.1-d1 |
1369.1-d |
$2$ |
$2$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
1369.1 |
\( 37^{2} \) |
\( 37^{2} \) |
$2.49086$ |
$(a+6), (a-7)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$4$ |
\( 1 \) |
$1$ |
$15.27090535$ |
3.332384748 |
\( \frac{9261}{37} \) |
\( \bigl[a + 1\) , \( 0\) , \( a + 1\) , \( -a\) , \( -a + 1\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-a{x}-a+1$ |
| 1369.1-d2 |
1369.1-d |
$2$ |
$2$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
1369.1 |
\( 37^{2} \) |
\( 37^{4} \) |
$2.49086$ |
$(a+6), (a-7)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$15.27090535$ |
3.332384748 |
\( \frac{10503459}{1369} \) |
\( \bigl[a + 1\) , \( 0\) , \( a + 1\) , \( 4 a - 15\) , \( -8 a + 19\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(4a-15\right){x}-8a+19$ |
| 1369.1-e1 |
1369.1-e |
$1$ |
$1$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
1369.1 |
\( 37^{2} \) |
\( 37^{2} \) |
$2.49086$ |
$(a+6), (a-7)$ |
$2$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
|
|
$1$ |
\( 1 \) |
$0.088048490$ |
$35.84317866$ |
2.754728324 |
\( \frac{110592}{37} \) |
\( \bigl[0\) , \( 0\) , \( 1\) , \( -1\) , \( 0\bigr] \) |
${y}^2+{y}={x}^{3}-{x}$ |
| 1369.1-f1 |
1369.1-f |
$2$ |
$2$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
1369.1 |
\( 37^{2} \) |
\( 37^{2} \) |
$2.49086$ |
$(a+6), (a-7)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$4$ |
\( 1 \) |
$1$ |
$15.27090535$ |
3.332384748 |
\( \frac{9261}{37} \) |
\( \bigl[a\) , \( -a + 1\) , \( a\) , \( -a + 1\) , \( 1\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-a+1\right){x}+1$ |
| 1369.1-f2 |
1369.1-f |
$2$ |
$2$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
1369.1 |
\( 37^{2} \) |
\( 37^{4} \) |
$2.49086$ |
$(a+6), (a-7)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$15.27090535$ |
3.332384748 |
\( \frac{10503459}{1369} \) |
\( \bigl[a\) , \( -a + 1\) , \( a\) , \( -6 a - 9\) , \( 7 a + 12\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-6a-9\right){x}+7a+12$ |
*The rank, regulator and analytic order of Ш are
not known for all curves in the database; curves for which these are
unknown will not appear in searches specifying one of these
quantities.
